I would like to ask recommendations on what I should focus my time on. I have just admiration for the vast world of mathematics and I would like to learn as much as I possibly can. However, I would like to prioritize the more relevant topics first. My work revolves around financial mathematics. In particular, I would like to specialize in (1) Stochastic Analysis, (2) Probability Theory, and (3) Optimization.
My current background includes: (1) Real Analysis, (2) Measure Theory, (3) Abstract Algebra, (4) Differential Geometry, (5) Algebraic Geometry, (6) Basics of Probability Theory. I have completed all of Folland so I have elementary knowledge of Functional and Fourier Analysis. My probability knowledge is from completing Williams. I do not want to gain knowledge for employability, instead I would like to one day be able to produce meaningful research in financial mathematics.
Now, I am not sure to what extent I would require other topics for my goal of gaining a deep understanding of financial mathematics. As of right now, I was contemplating going through Rudin's book on Functional Analysis.(Question 1.) Would a cover-to-cover read prove worthwhile? Meaning, is such knowledge on Functional Analysis needed in my field or would my background from Folland be sufficient? I would most definitely finish it one day but for now I would like to focus on any and everything that has direct application to finance going through the most relevant material first.(Question 2.) Moreover, what about fields like Harmonic and Microlocal Analysis? Should learning such topics be of priority to someone trying to specialize in financial math?
From what information is available online, I have read that knowledge of Partial Differential Equations is particularly useful.(Question 3.) Is it useful to go into depth or would a shallow background suffice? If you could mention some textbooks on the level that would prove most efficient for my time, I would be extremely grateful.
As per what I have currently planned, I would simply be devoting my time towards getting through a text on Stochastic Calculus (Karatzas and Shreve) and one on Stochastic Differential Equations (Oksendal).
I have three years to kill before I finish my undergraduate program. Till then, I have no responsibility and am as free as a bird. I feel like that much time would suffice to get a thorough understanding of financial mathematics. Finishing my question, I would like to restate that I do not want to gain knowledge for employability, instead I would like to one day be able to produce meaningful research in financial mathematics. I would, again, be extremely grateful if someone could answer my questions.